MAT 367. Applied Mathematics Transforms ProjectDr. R. L. Herman Spring 2016

Computing Transforms

Introduction

In this project you will use computing environments for computing Fourier and Laplace transforms. In alater part of the project you will learn how to use discrete Fourier transforms to determine the frequencycontent of signals.

You will keep track of your observations with your partner and submit the results on the last class dayin report form. This project will count as your computer lab/project grade for the course. All work shouldbe typed with double-spacing and 12 pt font. You will be expected to use correct English grammar andpunctuation. This is a report and thus you will use proper sentence and paragraph formatting. You willbe graded on the evidence of work, mathematical detail and understanding, proper exposition and neatness.Your work should also be supported with properly labeled and embedded plots and properly numbered andpunctuated equations. Any references used should be cited as well. These reports will count towards theproject component of your grade.

1 Symbolic MATLAB

For those not familiar with MATLAB, there are some additional help files listed at the project site http:

//people.uncw.edu/hermanr/mat367/ProjectS16.htm. You can access MATLAB through Tealware, athttps://tealware.uncw.edu. MATLAB is under the Mathematics & Statistics folder, MATLAB2015a. Openthe program. The m-files provided on the project web page, or later in this document, can either be savedto the default MATLAB directory, or the text can be copied and pasted into the editor and then saved withthe original file name. Data and other files should be copied over to the same directory. You should see yourfiles in the Current Folder panel on the left of the screen. [See the Figure 1.] You can add your own folder,but be sure to open the folder before running the m-file.

The m-files are provided which can be copied into an editor, saved with a .m extension into the workingdirectory, and called in MATLAB by entering the file name (without the .m) in the Command Window.Many of these can also be run under GNU Octave. More about GNU Octave can be found in the nextsection.

In the next section we will investigate the use of symbolic computations in MATLAB to compute Fourierand Laplace transforms. In order to do this, we make use of the Symbolic Toolbox commands.

In order to declare variables as symbolic, you can enter the command

syms a b c x

Now enter the quadratic function,

Figure 1: MATLAB workspace showing the Command window in the lower center area and the file structureon the left side.

f = a*x^2 + b*x + c;

You can now differentiate and integrate f with respect to x:

diff(x,x)

int(f,x)

You can even solve ax2 + bx+ c = 0 by

solve(f,x)

The output is

ans =

-(b + (b^2 - 4*a*c)^(1/2))/(2*a)

-(b - (b^2 - 4*a*c)^(1/2))/(2*a)

You would need to write the answer in standard mathematical form and not in this output format. Thatmeans not using *, such as

x =−b±

√b2 − 4ac

2a.

Before proceeding, you should clear the MATLAB workspace by entering

clear

2

Fourier and Laplace Transforms

In the Command window you can also enter MATLAB commands one line at a time or you can save severallines of code to an m-file to be run separately. For example, enter the line

syms t w

to say that t and w are symbolic variables. Then enter the function f(t) = e−|t| as

f=exp(-abs(t))

Now, compute the Fourier transform:

F=fourier(f);

Here MATLAB computes

F(w) = c*int(f(x)*exp(s*i*w*x),x,-inf,inf)

The constant c can be set, but you need not d that. It automatically gives a function of ω.

The inverse Fourier transform can be computed as

ifourier(2/(w^2 + 1))

If you want to do more, like plotting, you can use the ezplot command to plot symbolic functions. Awindow is provided by inserting [xmin xmax ymin ymax] as an argument.

The plot of f(t) is obtained using

ezplot(f,[-10,10,0,1 ])

or a plot of the Fourier transform is found using

ezplot(abs(F),[-20,20,0,2])

Next we use these commands to build an m-file and embellish the plots. Type, or copy, the text in thecode below in the Editor. Save the file as ftrans.m to the working directory. Then all you need to type isftrans in the Command window. [Note: Whenever copying and pasting code with quotes, you might needto retype the quotes.]

• ftrans.m

In the following code the function fourier computes the Fourier transform symbolically. The rest of

the code is used to plot f(t) = e−|t| and its transform, F =2

ω2 + 1. The result is shown in Figure 2.

3

t

-10 -5 0 5 10

f(t)

0

0.5

1f(t) vs t

ω

-20 -10 0 10 20

F(ω

)0

1

2Fourier Transform of f(t)

Figure 2: Plot of f(t) = e−|t| and its transform.

clear

syms t w

f=exp(-abs(t));

F=fourier(f);

figure(1)

subplot(2,1,1)

ezplot(f,[-10,10,0,1 ])

xlabel(’t’)

ylabel(’f(t)’)

title([’f(t) vs t’] )

subplot(2,1,2)

ezplot(abs(F),[-20,20,0,2])

xlabel(’\omega’)

ylabel(’F(\omega)’)

title([’Fourier Transform of f(t)’] )

One can also perform Laplace transforms,

syms x y

f = 1/sqrt(x);

laplace(f, x, y)

and inverse Laplace transforms

syms x y

F = 1/y^2;

ilaplace(F, y, x)

4

Convolution

Recall, the convolution is defined as

(f ∗ g)(x) =

∫ ∞−∞

f(ξ)g(x− ξ) dξ. (1)

We can compute the convolution directly by defining the functions f(x) and g(x) and carrying out theintegration in MATLAB.

Consider the text example. We compute the convolution of the box function

f(x) =

{1, |x| ≤ 1,0, |x| > 1,

and the half-triangular function

g(x) =

{x, 0 ≤ x ≤ 1,0, otherwise,

as shown in Figure 3.

x

f(x)

1−1

1

x

g(x)

1−1

1

Figure 3: A plot of the box function f(x) and the triangle function g(x).

We begin by defining these functions and then computing the convolution. We have to make use of theHeaviside step function,

H(x) =

{0, x < 0,1, x > 0.

(2)

The Heaviside function can be used to create piecewise-defined functions. Thus, the box function is givenby

f(x) = H(x+ 1)−H(x− 1)

and the half-triangular function isg(x) = x(H(x)−H(x− 1)).

Now, we turn to MATLAB. Enter the lines

5

clear

syms x xi

Next, enter the functions

f=(heaviside(x+1)-heaviside(x-1));

g=x*(heaviside(x)-heaviside(x-1));

In order to compute the convolution, as given in Equation (1), we need to write f and g as functions ofthe integration variable. We do this with the next lines.

g2=subs(g,x,x-xi);

f2=subs(f,x,xi);

Here we replaced the symbolic variable, x, with either ξ or x− ξ.

We are now ready to carry out the integration.

h=int(f2*g2,xi,[-10 10]);

Typing h in the Command window, we obtain the convolution in the form

piecewise([10 <= x, 0], [x <= 10, (heaviside(x + 1)*(x + 1)^2)/2

- (heaviside(x - 1)*(x - 1)^2)/2 - (x*heaviside(x)*(x + 2))/2

+ (x*heaviside(x - 2)*(x - 2))/2])

This is a bit unweildy. So, a plot would be nice. Below we provide an m-file which carries out the aboveconvolution and plots the results. The plots are shown in Figure 4. Note that the plot of the convolutionagrees with the text example.

There are commented out lines, denoted with the %. One can remove the % to try other convolutions,such as convolving a function with itself or using the full-triangular function that is provided.

• myconv.m

% Compute the convolution of f and g

clear

close all

syms x xi

% Enter f and g

f=(heaviside(x+1)-heaviside(x-1)); % box function

g=x*(heaviside(x)-heaviside(x-1)); % half triangle

% full triangle

6

%g=(x+1)*(heaviside(x+1)-heaviside(x))+(1-x)*(heaviside(x)-heaviside(x-1));

%g=f;

%f=g;

% Compute convolution integral

g2=subs(g,x,x-xi); % g(x-xi)

f2=subs(f,x,xi); % f(xi)

h=int(f2*g2,xi,[-10 10]); % Set large integration range

% Set x values for plotting in t variable

dt = 0.01;

t = [ -5:dt:5 ];

% Change symbolic functions to vectors for plotting

f1=subs(f,x,t);

f1=double(f1);

g1=subs(g,x,t);

g1=double(g1);

%Plot functions

figure(1)

subplot(3,1,1)

plot(t,f1)

xlabel(’x’)

ylabel(’f(x)’)

title(’f(x) vs x’)

subplot(3,1,2)

plot(t,g1)

xlabel(’x’)

ylabel(’g(x)’)

title(’g(x) vs x’)

h1=subs(h,x,t);

h1=double(h1);

subplot(3,1,3)

plot(t,h1)

axis([-5 5 0 1])

xlabel(’x’)

ylabel(’(f * g)(x)’)

title(’(f * g)(x) vs x’)

7

x

-5 0 5

f(x)

00.5

1f(x) vs x

x

-5 0 5

g(x)

00.5

1g(x) vs x

x

-5 0 5

(f

* g

)(x)

00.5

1(f * g)(x) vs x

Figure 4: Plots of the box and half-triangular functions and the convolution.

2 Problems

Here are some problems to try in MATLAB. For more complicated output, try using the simplify command.

1. For each function, find the Fourier transform and plot the function and the magnitude of its transform.Make sure to plot on large enough interval to capture the function.

a. f(t) = t2e−|t|.

b. f(t) = e−t2+2t.

c. f(t) = H(t+ 1)−H(t− 1).

d. f(t) = (t+ 1)(H(t+ 1)−H(t)) + (1− t)(H(t)−H(t− 1)).

e. f(t) = (H(t+ π)−H(t− π)) cos 5πt.

2. For each function, find the inverse Fourier transform.

a. f̂(ω) =2

ω2 + 9.

b. f̂(ω) =1

4 + iω2.

c. f̂(ω) =4

(ω2 + 4)2.

3. Consider the function f(t) and its Fourier transform, f̂(ω). Establish an uncertainty principle betweenthe spreads of the function and its transform: ∆t∆ω = const. Define the width of a function as the FullWidth at Half Maximum (FWHM).

8

a. f(t) = e−|t|

b. f(t) = e−at2

, a > 0.

4. Find the Laplace transform of the following functions:

a. f(t) = t2 − 5t+ 1.

b. f(t) = e3t−4.

c. f(t) = e3t sin 2t.

d. f(t) = t2H(t− 1).

e. f(t) =∫ t

0(t− u)2 sinu du.

5. Find the inverse Laplace transform of the following functions.

a. F (s) =9

s4+

7

s3.

b. F (s) =s+ 1

s2 + 1.

c. F (s) =3

s2 + 2s+ 2.

d. F (s) =1

(s− 1)2.

e. F (s) =e−3s

s2 − 1.

f. F (s) =1

s2 + 4s− 5.

6. Consider the following functions:

• the box function, f(x) = H(x+ 1)−H(x− 1)

• the full triangular function, h(x) = (1 + x)(H(x+ 1)−H(x)) + (1− x)(H(x)−H(x− 1)).

Find and plot the convolution of the following pairs of functions,

a. Box function with itself, (f ∗ f)(x).

b. Box function with full triangle function, (f ∗ h)(x).

b. The full triangle function with itself, (h ∗ h)(x).

3 Other Software

MATLAB is not the only software useful for carrying out these computations. We could use Maple, butsometimes open source alternatives are better. In this section we will discuss the use of GNU Octave andPython.

9

GNU Octave

GNU Octave is a MATLAB clone which is free and open. It can be obtained from https://www.gnu.org/

software/octave/. Many of the MATLAB m-files can be run in GNU Octave with little change. However,the symbolic package has to be installed.

Open GNU Octave and type

pkg install -forge symbolic

This need only been done one time.

Before running any m-files containing symbolic computation, you have to invoke the command

pkg load symbolic

Python

One can use Python to carry out similar computations as one can do in MATLAB. You can either install aPython interface like Anaconda at https://www.continuum.io/downloads or use online Python sites suchas found at https://www.getdatajoy.com or http://live.sympy.org/.

• ftrans.py

from sympy import *

x, k = symbols(’x k’)

F=fourier_transform(exp(-x**2), x, k)

print F

plot(F, (k, -1, 1), ylim=(0, 2))

• ltrans.py

from sympy import *

t, s = symbols(’t s’)

Y=laplace_transform(t**2, t, s)

print Y

4 Simulink

Simulink is a graphical environment for designing simulations of systems. When you have access to Simulinkand MATLAB, you can start MATLAB and on the icon bar there is an icon that you can click to launchSimulink. Alternatively, you can type to bring up the Simulink Library Browser as shown in Figure 5. Next,click the yellow plus to bring up a new model. We build models by dragging and connecting the neededcomponents, or blocks, from groups such as the Continuous, Math Operations, Sinks, or Sources.

10

Figure 5: The Simulink Library Browser. This is where various blocks can be found for constructing models.[As seen in MATLAB 2015a.]

5 Discrete Fourier Transform

The goal of this part of the project is to use MATLAB to investigate the use of the Discrete Fourier Transform(DFT) for the spectral analysis of time series. Several files will be needed in this project. They are providedon the project webpage.

MATLAB for the Discrete Fourier Transform

In this section we provide implementations of the discrete trigonometric transform in MATLAB. The firstimplementation is a straightforward one which can be done in most programming languages. The secondimplementation makes use of matrix computations that can be performed in MATLAB or similar programslike GNU Octave or Pylab. Sums can be done with matrix multiplication, as described in the next section.This eliminates the loops in the first program below and speeds up the computation for large data sets.

Direct Implementation for a data set

11

The following code was used to produce results for a data set. It shows a direct implementation usingloops to compute the trigonometric DFT as developed in this chapter. The data is entered in vector y.The Fourier coefficients are entered using the matrix capabilities of MATLAB as described in the nextsection. The signal is then reconstructed using the finite series representation. Plots are provided to showthis implementations as demonstrated in earlier examples.

%

% DFT in a direct implementation

%

% Enter Data in y

y=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 10.1 ...

9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 13.5 11.4 10.9 8.1];

% Get length of data vector or number of samples

N=length(y);

% Compute Fourier Coefficients

for p=1:N/2+1

A(p)=0;

B(p)=0;

for n=1:N

A(p)=A(p)+2/N*y(n)*cos(2*pi*(p-1)*n/N)’;

B(p)=B(p)+2/N*y(n)*sin(2*pi*(p-1)*n/N)’;

end

end

A(N/2+1)=A(N/2+1)/2;

% Reconstruct Signal - pmax is number of frequencies used

% in increasing order

pmax=13; for n=1:N

ynew(n)=A(1)/2;

for p=2:pmax

ynew(n)=ynew(n)+A(p)*cos(2*pi*(p-1)*n/N)+B(p) ...

*sin(2*pi*(p-1)*n/N);

end

end

% Plot Data

plot(y,’o’)

% Plot reconstruction over data

hold on

plot(ynew,’r’)

hold off

title(’Reconstruction of Monthly

Mean Surface Temperature’)

xlabel(’Month’)

ylabel(’Temperature’)

Direct Implementation for a function.

The next routine shows how we can determine the spectral content of a signal, given in this case by a

12

function and not a measured time series. The output is the original data and reconstructed Fourier seriesin Figure 1, the trigonometric DFT coefficients in Figure 2, and the the power spectrum in Figure 3. Thiscode will be referred to as ftex.m.

% ftex.m

% IMPLEMENTATION OF DFT USING TRIGONOMETRIC FORM

% N = Number of samples

% T = Record length in time

% y = Sampled signal

%

clear

N=128;

T=5;

dt=T/N;

t=(1:N)*dt;

f0=30;

y=sin(2*pi*f0*t);

% Compute arguments of trigonometric functions

for n=1:N

for p=0:N/2

Phi(p+1,n)=2*pi*p*n/N;

end

end

% Compute Fourier Coefficients

for p=1:N/2+1

A(p)=2/N*y*cos(Phi(p,:))’;

B(p)=2/N*y*sin(Phi(p,:))’;

end

A(1)=2/N*sum(y);

A(N/2+1)=A(N/2+1)/2;

B(N/2+1)=0;

% Reconstruct Signal - pmax is number of frequencies

% used in increasing order

pmax=N/2;

for n=1:N

ynew(n)=A(1)/2;

for p=2:pmax

ynew(n)=ynew(n)+A(p)*cos(Phi(p,n))+B(p)*sin(Phi(p,n));

end

end

% Plot Data

figure(1)

plot(t,y,’o’)

13

% Plot reconstruction over data

hold on

plot(t,ynew,’r’)

xlabel(’Time’)

ylabel(’Signal Height’)

title(’Reconstructed Signal’)

hold off

% Compute Frequencies

n2=N/2;

f=(0:n2)/(n2*2*dt);

% Plot Fourier Coefficients

figure(2)

subplot(2,1,1)

stem(f,A)

xlabel(’Frequency’)

ylabel(’Amplitude’)

title(’A’)

subplot(2,1,2)

stem(f,B)

xlabel(’Frequency’)

ylabel(’Amplitude’)

title(’B’)

% Plot Fourier Spectrum

figure(3)

Power=sqrt(A.^2+B.^2);

stem(f,Power(1:n2+1))

xlabel(’Frequency’)

ylabel(’Power’)

title(’Periodogram’)

% Show Figure 1

figure(1)

Compact Implementation - without loops.

The next implementation uses matrix products to eliminate the for loops in the previous code. The waythis works is described in the next section.

%

% DFT in a compact implementation

%

% Enter Data in y

y=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 ...

14

10.1 9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 ...

13.5 11.4 10.9 8.1];

N=length(y);

% Compute the matrices of trigonometric functions

p=1:N/2+1;

n=1:N;

C=cos(2*pi*n’*(p-1)/N);

S=sin(2*pi*n’*(p-1)/N);

% Compute Fourier Coefficients

A=2/N*y*C;

B=2/N*y*S;

A(N/2+1)=A(N/2+1)/2;

% Reconstruct Signal - pmax is number of frequencies used

% in increasing order

pmax=13;

ynew=A(1)/2+C(:,2:pmax)*A(2:pmax)’+S(:,2:pmax)*B(2:pmax)’;

% Plot Data

plot(y,’o’)

% Plot reconstruction over data

hold on

plot(ynew,’r’)

hold off

title(’Reconstruction of Monthly

Mean Surface Temperature’)

xlabel(’Month’)

ylabel(’Temperature’)

MATLAB Implementation of FFT

In this section we provide implementations of the Fast Fourier Transform in MATLAB. The MATLAB codeprovided in Section 5 can be simplified by using the built-in fft function. The following code employsMATLAB’s built-in fft function to replace the direct implementation code.

Implementation of the fft function.

% fanal.m

%

% Analysis Using FFT

%

clear

n=128;

15

T=5;

dt=T/n;

f0=6.2;

f1=10;

y=sin(2*pi*f0*(1:n)*dt)+2*sin(2*pi*f1*(1:n)*dt);

% y=exp(-0*(1:n)*dt).*sin(2*pi*f0*(1:n)*dt);

Y=fft(y,n);

n2=n/2;

Power=Y.*conj(Y)/n^2;

f=(0:n2)/(n2*2*dt);

stem(f,2*Power(1:n2+1))

xlabel(’Frequency’)

ylabel(’Power’)

title(’Periodogram’)

figure(2)

subplot(2,1,1)

stem(f,real(Y(1:n2+1)))

xlabel(’Frequency’)

ylabel(’Amplitude’)

title(’A’)

subplot(2,1,2)

stem(f,imag(Y(1:n2+1)))

xlabel(’Frequency’)

ylabel(’Amplitude’)

title(’B’)

figure(1)

This code can be simplified even further by placing the FFT computation and plot functions into afunction that accepts the signal as input. Below is the function fanalf, which can make the main programmore compact.

Putting fft and plots into a stand alone function.

function z=fanalf(y,T)

%

% FFT Analysis Function

%

% Enter Data in y and record length T

% Example:

% n=128;

% T=5;

% dt=T/n;

% f0=6.2;

% f1=10;

% fanal(sin(2*pi*f0*(1:n)*dt)+2*sin(2*pi*f1*(1:n)*dt));

% or

16

% fanal(sin(2*pi*6.2*(1:128)/128*5),5);

n=length(y);

dt=T/n;

Y=fft(y,n);

n2=floor(n/2);

Power=Y.*conj(Y)/n^2;

f=(0:n2)/(n2*2*dt);

z=Power;

stem(f,2*Power(1:n2+1))

xlabel(’Frequency’)

ylabel(’Power’)

title(’Periodogram’)

Examples of the use of this function for doing a spectral analysis of functions, data sets, and sound filesare provided below. This code snippet can be used with fanalf to analyze a given function.

Code for analysis of functions.

% fanal2.m

clear

n=128;

T=5;

dt=T/n;

f0=6.2;

f1=10;

t=(1:n)*dt;

y=sin(2*pi*f0*t)+2*sin(2*pi*f1*t);

fanalf(y,T);

The code snippet that can be used with fanalf to analyze a data set inserted in vector y is given by thenext set of code.

Code for analysis of typed data values.

y=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 ...

10.1 9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 ...

13.5 11.4 10.9 8.1];

n=length(y);

y=y-mean(y);

T=24;

dt=T/n;

fanalf(y,T);

One can also enter data stored in an ASCII file. The following code shows how this is done.

17

0 0.2 0.4 0.6 0.8 1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Figure 6: Typical signal from a sound file.

Code for analysis of data read from ASCII file.

[year,y]=textread(’sunspot.txt’,’%d %f’);

n=length(y);

t=year-year(1);

y=y-mean(y);

T=t(n);

dt=T/n;

fanalf(y’,T);

The next code snippet can be used with fanalf to analyze a given sound stored in a wav file. As anexample, Figure 6 shows the signal from the ringing of an old-style phone. The frequency content is displayedin Figure 7. We should note that the waveread function will be superceded by audioread in MATLAB.

Code for analysis of sound files.

[y,NS,NBITS]=wavread(’sound1.wav’);

n=length(y);

T=n/NS

dt=T/n;

figure(1)

plot(dt*(1:n),y)

figure(2)

fanalf(y,T);

18

Frequency

0 1000 2000 3000 4000 5000

Po

wer

×10-3

1

2

3

4

5

6

Periodogram

Figure 7: Spectral content of the sound in Figure 6.

Problems

1. The goal of this problem is to use MATLAB to investigate the use of the Discrete Fourier Transform(DFT) for the spectral analysis of time series. Running the m-files is done by typing the filename without theextension (.m) after the prompt (¿¿) in the Command Window, which is the middle panel of the MATLABprogram. Note, most of the code provided can be run in GNU Octave.

1. Analysis of simple functions.

a. File ftex.m - See the MATLAB section for the code.This file is a MATLAB program implementing the discrete Fourier transform using trigono-metric functions like that derived in the text. The input is a function, sometimes withdifferent frequencies. The output is a plot of the data points and the function fit, the Fouriercoefficients and the periodogram giving the power spectrum.

i. Save the file ftex.m in your working directory under MATLAB.

ii. View the file by entering edit ftex. Note how the first function is defined by the variabley.

iii. Run the file by typing ftex in MATLAB’s Command window.

iv. Change the parameters in ftex.m, remembering the original ones. In particular, changethe number of points, N, (keeping them even), the frequency, f0, and the record length,L. Note the effects. If you get an error, enter clear and try again. Always save the m-fileafter making any changes before running ftex.

v. Reset the parameters to the original values. What happens for frequencies of f0 = 5, 5.5and 5.6? Is this what you expected?

vi. Repeat the last set of frequencies for double the record length, T. Is there a change?

vii. Reset the parameters. Put in frequencies of f0 = 20, 30, 40. What frequencies are presentin the periodogram? Is this what you expected?

b. Look at sums of several trigonometric functions.

19

i. Reset the parameters.

ii. Change the function in y to y=sin(2*pi*f0*t)+sin(2*pi*f1*t); and add a line defin-ing f1. Start with f1 = 3; and look at several other values for the two frequencies. Trydifferent amplitudes; for example, 3*sin(2*pi*f0*t) has an amplitude of 3. Recordyour observations.

iii. Change one of the sines to a cosine. What is the effect? What happens when the sineand cosine terms have the same frequency?

c. Investigate non-sinusoidal functions.

i. Investigate the following functions:

1. y=t;

2. y=t.^2;

3. y=sin(2*pi*f0*(t-T/5))./(t-T/5); What is this function?

4. y(1,1:M)=ones(1,M); y(1,M+1:N)=zeros(1,N-M); Start with M = N/2; What isthis function? How are the last two problems related? Do they relate to anythingfrom earlier class lectures? What effect results from changing M?

5. Try multiplying the function in 4 by a simple sinusoid; for example, add the liney=sin(2*pi*f0*t).*y, for M = N/2. How does this affect what you had gotten forthe sinusoid without multiplication?

2. Use the FFT function.

In MATLAB there is a built in set of functions, fft and ifft for the computation of the DiscreteExponential Transform and its inverse using the Fast Fourier Transform (FFT). The files neededto do this are fanal.m or fanal2.m and fanalf.m. [See Section 5 for the code.] Put these codesinto the MATLAB editor and see what they look like. Note that fanal.m was split into the twofiles fanal2.m and fanalf.m. This will allow us to be confident when we later create new datafiles and then using fanalf.m. Test this set of functions for simple sine functions to see that youget results similar to Part 1. First run fanal and then fanal2. Is there any difference betweenthese last two approaches?

3. Analysis of data sets.

One often does not have a function to analyze. Some measurements are made over time of acertain quantity. This is called a time series. It could be a set of data describing things likethe stock market fluctuations, sunspots activity, ocean wave heights, etc. Large sets of data canbe read into y and small sets can be input as vectors. We will look at how this can be done.After the data is entered, one can analyze the time series to look for any periodic behavior, or itsfrequency content. In the two cases below, make sure you look at the original time series usingplot(y,+).

a. Ocean WavesIn this example the data consists of monthly mean sea surface temperatures (oC) at onepoint over a 2 year period. The temperatures are placed in y as a row vector. Note how thedata is continued to a second line through the use of ellipsis. Also, one typically subtracts theaverage from the data. What affect should this have on the spectrum? Copy the followingcode into a new m-file called fdata.m and run fdata. Determine the dominant periods inthe monthly mean sea surface temperature.

y=[7.6 7.4 8.2 9.2 10.2 11.5 12.4 13.4 13.7 11.8 ...

10.1 9.0 8.9 9.5 10.6 11.4 12.9 12.7 13.9 14.2 ...

13.5 11.4 10.9 8.1];

20

n=length(y);

y=y-mean(y);

T=24;

dt=T/n;

fanalf(y,T);

b. SunspotsSunspot data was exported to a text file. Download the file sunspot.txt. You can open thedata in Notepad and see the two column format. The times are given as years (like 1850).This example shows how a time series can be read and analyzed. From your spectrum,determine the major period of sunspot activity. Note that we first subtracted the averageso as not to have a spike at zero frequency. Copy and paste into the editor and save asfanaltxt.m. Note: Copy and Paste of single quotes often does not work correctly. Retypethe single quotes after copying.

[year,y]=textread(’sunspot.txt’,’%d %f’);

n=length(y);

t=year-year(1);

y=y-mean(y);

T=t(n);

dt=T/n;

fanalf(y’,T);

4. Analysis of sounds. Sounds can be input into MATLAB. You can create your own sounds inMATLAB or sound editing programs like Audacity or Goldwave to create audio files. These filescan be input into MATLAB for analysis.

Save the following sample code as fanalwav.m and save the first sound file. This code showshow one can read in a WAV file. There will be two plots, the first showing the wave profile andthe second giving the spectrum. Try some of the other wav files and report your findings. Note:Copy and Paste of single quotes often does not work correctly. Retype the single quotes aftercopying.

[y,NS,NBITS]=wavread(’sound1.wav’);

n=length(y);

T=n/NS

dt=T/n;

figure(1)

plot(dt*(1:n),y)

figure(2)

fanalf(y,T);

Several wav files are be provided for you to analyze. If you are able to hear the sounds, you canrun them from MATLAB by typing sound(y,NS). In fact, you can even create your own soundsbased upon simple functions and save them as wav files. For example, try the following code:Note: Copy and Paste of single quotes often does not work correctly. Retype the single quotesafter copying.

smp=11025;

t=(1:2000)/smp;

y=0.75*sin(2*pi*440*t);

sound(y,smp,8);

wavwrite(y,smp,8, ’myfile.wav’);

21

Try some other functions, using several frequencies. If you get a clipping error, then reduce theamplitudes you are using.

22